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- Posterior_predictive_distribution abstract "In statistics, and especially Bayesian statistics, the posterior predictive distribution is the distribution that a new i.i.d. data point would have, given a set of N existing i.i.d. observations . In a frequentist context, this might be derived by computing the maximum likelihood estimate (or some other estimate) of the parameter(s) given the observed data, and then plugging them into the distribution function of the new observations.However, the concept of posterior predictive distribution is normally used in a Bayesian context, where it makes use of the entire posterior distribution of the parameter(s) given the observed data to yield a probability distribution over an interval rather than simply a point estimate. Specifically, it is computed by marginalising over the parameters, using the posterior distribution:where represents the parameter(s) and the hyperparameter(s). Any of may be vectors (or equivalently, may stand for multiple parameters).Note that this is equivalent to the expected value of the distribution of the new data point, when the expectation is taken over the posterior distribution, i.e.:(To get an intuition for this, keep in mind that expected value is a type of average. The predictive probability of seeing a particular value of a new observation will vary depending on the parameters of the distribution of the observation. In this case, we don't know the exact value of the parameters, but we have a posterior distribution over them, that specifies what we believe the parameters to be, given the data we've already seen. Logically, then, to get "the" predictive probability, we should average all of the various predictive probabilities over the different possible parameter values, weighting them according to how strongly we believe in them. This is exactly what this expected value does. Compare this to the approach in frequentist statistics, where a single estimate of the parameters, e.g. a maximum likelihood estimate, would be computed, and this value plugged in. This is equivalent to averaging over a posterior distribution with no variance, i.e. where we are completely certain of the parameter having a single value. The result is weighted too strongly towards the mode of the posterior, and takes no account of other possible values, unlike in the Bayesian approach.)".
- Posterior_predictive_distribution wikiPageID "35052447".
- Posterior_predictive_distribution wikiPageRevisionID "564261035".
- Posterior_predictive_distribution hasPhotoCollection Posterior_predictive_distribution.
- Posterior_predictive_distribution subject Category:Bayesian_statistics.
- Posterior_predictive_distribution subject Category:Probability_theory.
- Posterior_predictive_distribution comment "In statistics, and especially Bayesian statistics, the posterior predictive distribution is the distribution that a new i.i.d. data point would have, given a set of N existing i.i.d. observations .".
- Posterior_predictive_distribution label "Posterior predictive distribution".
- Posterior_predictive_distribution sameAs m.0j67hyc.
- Posterior_predictive_distribution sameAs Q7234227.
- Posterior_predictive_distribution sameAs Q7234227.
- Posterior_predictive_distribution wasDerivedFrom Posterior_predictive_distribution?oldid=564261035.
- Posterior_predictive_distribution isPrimaryTopicOf Posterior_predictive_distribution.